💡First, I need to determine the new edge length of the cube after increasing it by 15%. The original edge length is 7 cm. To find the increase, I'll calculate 15% of 7 cm and then add it to the original length.Next, I'll use the new edge length to calculate the new surface area of the cube. The surface area of a cube is given by the formula (6 times (text{edge length})^2). I'll substitute the new edge length into this formula to find the updated surface area.

💡Okay, so I have this problem about three plots of land with their areas in a certain ratio, and some information about the grain collected from them. I need to find the total area of all three plots. Let me try to break this down step by step.First, the problem says the areas are in the ratio (2 frac{3}{4}: 1 frac{5}{6}: 1 frac{3}{8}). Hmm, mixed numbers can be a bit tricky, so maybe I should convert these to improper fractions to make it easier to work with them.Let me do that:- (2 frac{3}{4}) is equal to (2 + frac{3}{4}). To convert this to an improper fraction, I multiply the whole number by the denominator and add the numerator: (2 times 4 = 8), plus 3 is 11, so it's (frac{11}{4}). - (1 frac{5}{6}) is (1 + frac{5}{6}). Similarly, (1 times 6 = 6), plus 5 is 11, so that's (frac{11}{6}). - (1 frac{3}{8}) is (1 + frac{3}{8}). That would be (1 times 8 = 8), plus 3 is 11, so (frac{11}{8}).So now, the ratios are (frac{11}{4} : frac{11}{6} : frac{11}{8}).Looking at these, I notice that each part of the ratio has 11 in the numerator. Maybe I can factor that out to simplify the ratios. If I divide each term by 11, the ratios become (frac{1}{4} : frac{1}{6} : frac{1}{8}).Now, these are still fractions, and it might be easier if I can express them as whole numbers. To do that, I can find a common denominator for the fractions (frac{1}{4}), (frac{1}{6}), and (frac{1}{8}). The least common denominator for 4, 6, and 8 is 24.So, converting each fraction to have a denominator of 24:- (frac{1}{4} = frac{6}{24})- (frac{1}{6} = frac{4}{24})- (frac{1}{8} = frac{3}{24})So now, the simplified ratio is (6:4:3). That makes it much easier to work with.Let me denote the areas of the three plots as (6k), (4k), and (3k) respectively, where (k) is a common multiplier.Next, the problem states that 72 quintals more grain were collected from the first plot than from the second. The average yield is 18 quintals per hectare. So, the amount of grain collected from each plot is the area multiplied by the yield.Therefore, the grain from the first plot is (6k times 18) quintals, and from the second plot, it's (4k times 18) quintals.The difference between these two amounts is 72 quintals. So, I can set up the equation:(6k times 18 - 4k times 18 = 72)Let me compute that:First, factor out the 18:(18 times (6k - 4k) = 72)Simplify inside the parentheses:(18 times 2k = 72)Which simplifies to:(36k = 72)Now, solving for (k):(k = frac{72}{36} = 2)Okay, so (k = 2). Now, I can find the areas of each plot:- First plot: (6k = 6 times 2 = 12) hectares- Second plot: (4k = 4 times 2 = 8) hectares- Third plot: (3k = 3 times 2 = 6) hectaresTo find the total area, I just add them up:(12 + 8 + 6 = 26) hectaresWait, let me double-check my calculations to make sure I didn't make a mistake. I converted the mixed numbers correctly to improper fractions, simplified the ratios, found the common denominator, set up the equation based on the grain difference, solved for (k), and then calculated each area. Everything seems to add up correctly. The total area is indeed 26 hectares.

💡First, I need to determine the total number of home runs scored by the Chicago Cubs during the game. They scored 2 home runs in the third inning, 1 home run in the fifth inning, and 2 more home runs in the eighth inning. Adding these together gives a total of 5 home runs for the Cubs.Next, I will calculate the total number of home runs scored by the Cardinals. They scored 1 home run in the second inning and 1 home run in the fifth inning, which adds up to 2 home runs in total.Finally, to find out how many more home runs the Chicago Cubs scored than the Cardinals, I subtract the Cardinals' total home runs from the Cubs' total home runs: 5 - 2 = 3 home runs.

💡First, I need to determine the total points Emily had before the last round by adding her scores from the first and second rounds.Emily scored 16 points in the first round and 33 points in the second round.So, the total points before the last round are:16 points (first round) + 33 points (second round) = 49 pointsAt the end of the game, Emily had 1 point. This means she lost the difference between the points she had before the last round and the points she had at the end.Therefore, the points lost in the last round are:49 points (before last round) - 1 point (end of the game) = 48 pointsThus, Emily lost 48 points in the last round.

💡First, I need to evaluate the expression (3(24 + 7)^2 - (24^2 + 7^2)).I'll start by simplifying the term inside the parentheses: (24 + 7 = 31).Next, I'll square this result: (31^2 = 961).Then, I'll multiply this squared value by 3: (3 times 961 = 2883).Now, I'll calculate the squares of 24 and 7 individually: (24^2 = 576) and (7^2 = 49).Adding these two results together gives: (576 + 49 = 625).Finally, I'll subtract the sum of the squares from the tripled squared value: (2883 - 625 = 2258).Therefore, the value of the expression is 2258.

💡Alright, so I've got this polar equation here: ρ = cos(π/4 - θ). And I need to figure out what kind of curve this represents. The options are hyperbola, ellipse, parabola, or circle. Hmm, okay.First, I remember that polar equations can sometimes be converted into Cartesian coordinates to make it easier to identify the shape. So maybe I should try that. I think the general conversion formulas are x = ρ cos θ and y = ρ sin θ, and ρ² = x² + y². Yeah, that sounds right.Looking at the equation ρ = cos(π/4 - θ), I wonder if I can simplify this using a trigonometric identity. I recall that cos(A - B) = cos A cos B + sin A sin B. So maybe I can expand cos(π/4 - θ) using that identity.Let's try that. So cos(π/4 - θ) = cos(π/4)cos θ + sin(π/4)sin θ. I know that cos(π/4) and sin(π/4) are both √2/2. So substituting those in, I get:cos(π/4 - θ) = (√2/2)cos θ + (√2/2)sin θ.So now the equation becomes ρ = (√2/2)(cos θ + sin θ). Hmm, okay.I think the next step is to multiply both sides by ρ to make it easier to convert to Cartesian coordinates. So multiplying both sides by ρ gives:ρ² = (√2/2)(ρ cos θ + ρ sin θ).Now, using the conversion formulas, ρ² is x² + y², and ρ cos θ is x, and ρ sin θ is y. So substituting those in, I get:x² + y² = (√2/2)(x + y).Hmm, that looks better. Now, to make it look more like a standard conic section equation, I might want to get rid of the fraction. So I can multiply both sides by 2/√2 to simplify. Let's see:Multiplying both sides by 2/√2:(2/√2)(x² + y²) = (2/√2)(√2/2)(x + y).Simplifying the right side, (2/√2)(√2/2) cancels out to 1, so we have:(2/√2)(x² + y²) = x + y.But 2/√2 is equal to √2, right? Because 2 divided by √2 is √2. So substituting that in:√2(x² + y²) = x + y.Okay, so now the equation is √2(x² + y²) = x + y. Hmm, this still looks a bit complicated, but maybe I can rearrange it to get all terms on one side.Let's subtract x and y from both sides:√2(x² + y²) - x - y = 0.Hmm, not sure if that helps. Maybe I should divide both sides by √2 to make the coefficients of x² and y² equal to 1. Let's try that:x² + y² - (x + y)/√2 = 0.Hmm, that's better. Now, to see if this is a circle, ellipse, parabola, or hyperbola, I might need to complete the square for both x and y terms.Let me rearrange the equation:x² - (x)/√2 + y² - (y)/√2 = 0.Okay, so grouping the x terms and y terms:(x² - (x)/√2) + (y² - (y)/√2) = 0.Now, to complete the square for the x terms: x² - (x)/√2. The coefficient of x is -1/√2, so half of that is -1/(2√2), and squaring that gives 1/(8). Similarly for the y terms: y² - (y)/√2. The coefficient of y is -1/√2, so half of that is -1/(2√2), and squaring that is also 1/(8).So, adding and subtracting 1/8 for both x and y:(x² - (x)/√2 + 1/8) - 1/8 + (y² - (y)/√2 + 1/8) - 1/8 = 0.Simplifying, this becomes:(x - 1/(2√2))² + (y - 1/(2√2))² - 1/4 = 0.Adding 1/4 to both sides:(x - 1/(2√2))² + (y - 1/(2√2))² = 1/4.Ah, okay! So this is the equation of a circle with center at (1/(2√2), 1/(2√2)) and radius 1/2. So the original polar equation represents a circle.Wait, but let me double-check my steps to make sure I didn't make a mistake. Starting from ρ = cos(π/4 - θ), I expanded it using the cosine difference identity, converted to Cartesian coordinates, rearranged, completed the square, and ended up with a circle equation. That seems correct.I think I did everything right. So the answer should be a circle, which is option D.